From 9aa887b6e9cf240de5efbd26411d6a2aad834366 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 15:18:53 +0100
Subject: Started converting a to r where relevant.

---
 bezout.tex | 83 ++++++++++++++++++++++++++++++++++----------------------------
 1 file changed, 46 insertions(+), 37 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index c0ecff7..eaa98ce 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -122,6 +122,18 @@ referenced above. The same cannot be said for the space defined by $|z|^2=N$,
 which is topologically the $(2N-1)$-sphere and cannot admit a complex
 structure.
 
+A consequence of the constraint is that the model's configuration space is not
+compact, nor is its energy bounded. This is not necessarily a problem, as many
+related problems have similar properties but are concerned with subspaces on
+which the energy is bounded. (In fact, identifying the appropriate subspace on
+which to define one's model often requires the study of critical points in the
+whole space.) Where it might be a problem, we introduce the additional
+constraint $|z|^2\leq Nr^2$. The resulting configuration space is a complex
+manifold with boundary. We shall see that the `radius' $r$ proves an insightful
+knob in our present problem, revealing structure as it is varied. Note
+that---combined with the constraint $z^2=N$---taking $r=1$ reduces the problem
+to that of the ordinary $p$-spin.
+
 The critical points are of $H$ given by the solutions to the set of equations
 \begin{equation} \label{eq:polynomial}
   \frac{p}{p!}\sum_{j_1\cdots j_{p-1}}^NJ_{ij_1\cdots j_{p-1}}z_{j_1}\cdots z_{j_{p-1}}
@@ -145,7 +157,7 @@ of $2N$ real variables. Its number of saddle-points is given by the usual
 Kac--Rice formula:
 \begin{equation} \label{eq:real.kac-rice}
   \begin{aligned}
-    \mathcal N_J&(\kappa,\epsilon)
+    \mathcal N_J&(\kappa,\epsilon,r)
       = \int dx\,dy\,\delta(\partial_x\operatorname{Re}H)\delta(\partial_y\operatorname{Re}H) \\
       &\hspace{6pc}\times\left|\det\begin{bmatrix}
           \partial_x\partial_x\operatorname{Re}H & \partial_x\partial_y\operatorname{Re}H \\
@@ -160,7 +172,7 @@ $\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying these
 transformations through, we have
 \begin{equation} \label{eq:complex.kac-rice}
   \begin{aligned}
-    \mathcal N_J&(\kappa,\epsilon)
+    \mathcal N_J&(\kappa,\epsilon,r)
       = \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H) \\
                 &\hspace{6pc}\times\left|\det\begin{bmatrix}
             \operatorname{Re}\partial\partial H & -\operatorname{Im}\partial\partial H \\
@@ -203,7 +215,7 @@ and auxiliary fields to eight bilinears defined by $Na=|z|^2$, $N\hat a=|\hat
 z|^2$, $N\hat c=\hat z^2$, $Nb=\hat z^*z$, and $Nd=\hat zz$ (and their
 conjugates). The result, to leading order in $N$, is
 \begin{equation} \label{eq:saddle}
-    \overline{\mathcal N}(\kappa,\epsilon)
+    \overline{\mathcal N}(\kappa,\epsilon,r)
         = \int da\,d\hat a\,db\,db^*d\hat c\,d\hat c^*dd\,dd^*e^{Nf(a,\hat a,b,\hat c,d)},
 \end{equation}
 where the argument of the exponential is
@@ -352,10 +364,11 @@ geometry problem, and yields
 for $\delta=\kappa a^{-(p-2)}$.
 
 Given $\rho$, the integral in \eqref{eq:free.energy.a} may be preformed for
-arbitrary $a$. The resulting expression is maximized for $a=\infty$ for all
-values of $\kappa$ and $\epsilon$. Taking this saddle gives
+arbitrary $a$. The resulting expression is maximized for $a=r^2$ for all
+values of $\kappa$ and $\epsilon$. Evaluating the complexity at this saddle, in
+the limit of unbounded spins, gives
 \begin{equation} \label{eq:bezout}
-  \log\overline{\mathcal N}(\kappa,\epsilon)
+  \lim_{r\to\infty}\log\overline{\mathcal N}(\kappa,\epsilon,r)
   =N\log(p-1).
 \end{equation}
 This is, to this order,  precisely the Bézout bound, the maximum number of
@@ -367,41 +380,34 @@ surprising, since the coefficients of our polynomial equations
 Analogous asymptotic scaling has been found for the number of pure Higgs states
 in supersymmetric quiver theories \cite{Manschot_2012_From}.
 
-More insight is gained by looking at the count as a function of $a$, defined by
-$\overline{\mathcal N}(\kappa,\epsilon,a)=e^{Nf(a)}$.  In the large-$N$ limit,
-this is the cumulative number of critical points, or the number of critical
-points $z$ for which $|z|^2\leq a$. We likewise define the $a$-dependant
-complexity $\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal
-N}(\kappa,\epsilon,a)$
-
 \begin{figure}[htpb]
   \centering
   \includegraphics{fig/complexity.pdf}
   \caption{
     The complexity of the 3-spin model at $\epsilon=0$ as a function of
-    $a=|z|^2=1+y^2$ at several values of $\kappa$. The dashed line shows
-    $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$.
+    the maximum `radius' $r=|z_{\mathrm{max}}|/\sqrt N$ at several values of
+    $\kappa$. The dashed line shows $\frac12\log(p-1)$, while the dotted shows
+    $\log(p-1)$.
   } \label{fig:complexity}
 \end{figure}
 
-Everything is analytically tractable for $\epsilon=0$, giving
+For finite $r$, everything is analytically tractable at $\epsilon=0$, giving
 \begin{equation} \label{eq:complexity.zero.energy}
-  \Sigma(\kappa,0,a)
-  =\log(p-1)-\frac12\log\left(\frac{1-|\kappa|^2a^{-2(p-1)}}{1-a^{-2}}\right).
+  \Sigma(\kappa,0,r)
+  =\log(p-1)-\frac12\log\left(\frac{1-|\kappa|^2r^{-4(p-1)}}{1-r^{-4}}\right).
 \end{equation}
-Notice that the limit of this expression as $a\to\infty$ corresponds with
-\eqref{eq:bezout}, as expected. This is plotted as a function of $a$ for
+This is plotted as a function of $r$ for
 several values of $\kappa$ in Fig.~\ref{fig:complexity}. For any $|\kappa|<1$,
-the complexity goes to negative infinity as $a\to1$, i.e., as the spins are
+the complexity goes to negative infinity as $r\to1$, i.e., as the spins are
 restricted to the reals.  This is natural, given that the $y$ contribution to
-the volume shrinks to zero as that of an $N$-dimensional sphere $\sum_i y_i^2=N(a-1)$ with volume
-$\sim(a-1)^N$.  However, when the result is analytically continued to
+the volume shrinks to zero as that of an $N$-dimensional sphere $\sum_i y_i^2=N(r^2-1)$ with volume
+$\sim(r^2-1)^N$.  However, when the result is analytically continued to
 $\kappa=1$ (which corresponds to real $J$) something novel occurs: the
-complexity has a finite value at $a=1$. Since the $a$-dependence gives a
+complexity has a finite value at $r=1$. Since the $r$-dependence gives a
 cumulative count, this implies a $\delta$-function density of critical points
 along the line $y=0$.  The number of critical points contained within is
 \begin{equation}
-  \lim_{a\to1}\lim_{\kappa\to1}\log\overline{\mathcal N}(\kappa,0,a)
+  \lim_{r\to1}\lim_{\kappa\to1}\log\overline{\mathcal N}(\kappa,0,r)
   = \frac12N\log(p-1),
 \end{equation}
 half of \eqref{eq:bezout} and corresponding precisely to the number of critical
@@ -414,27 +420,30 @@ $\epsilon$ is varied.
   \centering
   \includegraphics{fig/desert.pdf}
   \caption{
-    The minimum value of $a$ for which the complexity is positive as a function
-    of (real) energy $\epsilon$  for the 3-spin model at several values of
-    $\kappa$.
+    The value of `radius' $r$ for which $\Sigma(\kappa,\epsilon,r)=0$ as a
+    function of (real) energy per spin $\epsilon$  for the 3-spin model at
+    several values of $\kappa$. Above each line the complexity is positive and
+    critical points proliferate, while below it the complexity is negative and
+    critical points are exponentially suppressed. The dotted black lines show
+    the location of the ground and highest exited state energies for the real
+    3-spin model.
   } \label{fig:desert}
 \end{figure}
 
 In the thermodynamic limit \eqref{eq:complexity.zero.energy} implies that most
-critical points are concentrated at infinite $a$, i.e., at complex vectors with
-very large squared norm. For finite $N$ the expectation value $\langle
-a\rangle$ is likewise finite. By differentiating $\overline{\mathcal N}$ with
-respect to $a$ and normalizing, one has an approximation for the distribution
-of critical points as a function of $a$. The expectation value this yields is
-$\langle a\rangle\propto N^{1/2}+O(N^{-1/2})$. One therefore expects typical
+critical points are concentrated at infinite radius, i.e., at complex vectors with
+very large squared norm. For finite $N$ the average radius of critical points is likewise finite. By differentiating $\overline{\mathcal N}$ with
+respect to $r$ and normalizing, one has the distribution
+of critical points as a function of $r$. The average radius this yields is
+$\propto N^{1/4}+O(N^{-3/4})$. One therefore expects typical
 critical points to have a norm that grows modestly with system size.
 
 These qualitative features carry over to nonzero $\epsilon$. In
-Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $a$
+Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $r$
 close to one for which there are no solutions. When $\kappa=1$---the analytic
 continuation to the real computation---the situation is more interesting. In
 the range of energies where there are real solutions this gap closes, which is
-only possible if the density of solutions diverges at $a=1$.  Another
+only possible if the density of solutions diverges at $r=1$.  Another
 remarkable feature of this limit is that there is still a gap without solutions
 around `deep' real energies where there is no real solution. A moment's thought
 tells us that this is a necessity: otherwise a small perturbation of the $J$s
@@ -463,7 +472,7 @@ is richer than in the real case. In Fig.~\ref{fig:eggs} these are shown in the
 complex-$\epsilon$ plane for several examples. Depending on the parameters, the
 threshold might always come at smaller magnitude than the extremal state, or
 always come at larger magnitude, or cross as a function of complex argument.
-For sufficiently large $a$ the threshold always comes at larger magnitude than
+For sufficiently large $r$ the threshold always comes at larger magnitude than
 the extremal state. If this were to happen in the real case, it would likely
 imply our replica symmetric computation is unstable, since having a ground
 state above the threshold implies a ground state Hessian with many negative
-- 
cgit v1.2.3-54-g00ecf

